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Blow-ups of minimal surfaces in the Heisenberg group

  • Yonghao Yu EMAIL logo
Published/Copyright: March 14, 2025

Abstract

In this article, we revise Monti’s results on blow-ups of H-perimeter minimizing sets in Hn . Monti demonstrated that the Lipschitz approximation of the blow-up, after rescaling by the square root of the excess, converges to a limit function for n2 . However, the partial differential equation he derived for this limit function φ through contact variation is incorrect. Instead, the limit function solves the following equation weakly

y1Δ0φ=0,

MSC 2010: 040A30

1 Introduction

The study of geometric measure theory within the Heisenberg group Hn began with the monumental work [13]. Since then the literature on this topic has expanded substantially. Among the unresolved questions in this area, the regularity of perimeter-minimizing sets has attracted growing attention, as addressing this issue would be pivotal for solving the Heisenberg isoperimetric problem.

So far, most of the regularity results for H-minimal surfaces in Hn start with some a priori regularity [14].

However, the regularity theory of perimeter-minimizing sets in Rn does not need any initial regularity. It was developed by De Giorgi in the series of papers [68]. In Monti’s papers [11] and [12], he aimed to reproduce De Giorgi’s theory within the context of the Heisenberg group Hn . He succeeded in reproducing similar results in steps 1 and 2 but encountered difficulties in the third step. Below is a brief outline of De Giorgi’s theory and Monti’s approach:

The first step of De Giorgi’s theory is known as the Lipschitz approximation, which uses Lipschitz functions to approximate the perimeter minimizer. In Hn , Monti approximates the boundary of minimal surfaces using Lipschitz graphs. In the study by Monti [11], he demonstrated that the error of this approximation is bounded by a volume quantity times the excess of the boundary.

The second step in De Giorgi’s theory is the existence of a harmonic function. Monti [12] considered a H -perimeter minimizing set EHn in some neighborhood of 0Hn . By rescaling the set E with dilations δ1rh as rh goes to 0, Monti constructed a sequence of H -perimeter minimizing sets Eh with horizontal excess η2h approaching 0. Then, by using the Lipschitz approximation theorem [11], Monti obtained a sequence of intrinsic Lipschitz functions φh:DR that approximate the boundary of the rescaled sets Eh , where D is some open subset of the vertical hyperplane W={(x1,y1,,xn,yn,t)Hn:x1=0} . As stated in Theorem 3.1, for n2 , Monti showed that there exists a sequence of hi such that (φhηh) weakly converges to φ in L2(D) . He claimed that the limit function φ is independent of the first variable y1 . When E is strongly perimeter minimizing (Definition 2.3), he claimed that φ satisfied an equation involving the Kohn–Laplacian ΔH=ni=2X2i+Y2i . However, both claims are incorrect due to a calculation error. We correct his result in Theorem A.

We identify Hn with the set Cn×R by the coordinates (z,t) , where z=(x1+iy1,,xn+iyn) . Let Xi , Yi , i=1,,n denote the usual left-invariant vector fields defined on Hn , and Qr be the homogeneous cube centered at 0 with radius r , it is defined as follows:

Qr={(z,t)Hn:xi<r,yi<r,t<r2,i=1,,n},

Theorem A

For n2 , take any locally finite perimeter set EHn . Suppose 0*E and the horizontal inner normal νE(0)=(1,0,,0)Hn . Then:

  1. If E is H-perimeter minimizing in some neighborhood of 0Hn , the limit function φ in Theorem 3.1 solves the following equation weakly in D14 :

    (1.1) y1Δ0φ=0,

    where Δ0=2y21+ni=2X2i+Y2i , D14={(z,t)Q14:x1=0} .

  2. If E is strongly H-perimeter minimizing in some neighborhood of 0Hn , then the limit function φ solves the following equation weakly in D14 :

    (1.2) Δ0φ=0.

The third step is the decay estimate for excess, which can be shown if the function is harmonic. By iterating the excess decay inequality, one can show that the unit normal of the minimal set is Hölder continuous, which implies that the boundary of the minimal set is C1,α . However, it is unknown whether a similar excess decay lemma holds when the graphing function φ only satisfies a harmonic like PDE Δ0φ=0 . In the future, we plan to study this PDE to obtain similar regularity results of minimal surfaces on the Heisenberg group.

2 Preliminaries

The Heisenberg group Hn is the set Cn×R equipped with the group product,

(z,t)*(z,t)=(z+z,t+t2Imz,z),

where z,zCn , t,tR , and z,z=z1ˉz++znˉz .

The Lie algebra of Hn is spanned by the left-invariant vector fields,

Xk=xk+2ykt,Yk=yk2xkt,T=t,

where zk=xk+iyk , k=1,,n . The only nontrivial brackets are [Xk,Yk]=4T for k=1,,n .

The vector fields X1,Y1,,Xn,Yn are called the horizontal vector fields of Hn . We define H as the horizontal subbundle of THn , where:

H(p)

Let Ω H n be an open set. Take any continuous function V : Ω R 2 n . It can be identified with a horizontal vector field V = j = 1 n V j X j + V n + j Y j , then the horizontal divergence of V is

div H V = j = 1 n X j V j + Y j V n + j ,

In the Heisenberg group, one can define the H -perimeter similarly to the perimeter in Euclidean space.

Definition 2.1

( H -perimeter) Let E H n be a measurable subset and Ω H n be an open set. The H-perimeter of E in Ω is defined as follows:

P H ( E ; Ω ) = sup Ω χ E ( z , t ) div H V d z d t : V C 0 1 ( Ω ; R 2 n ) , V 1 ,

where χ E ( z , t ) denotes the characteristic function of E .

We say that E has finite H-perimeter in Ω if P H ( E ; Ω ) < . Moreover, E is said to have locally finite H-perimeter in Ω if for any open set A Ω , P H ( E ; A ) is finite. We denote P H ( E ; A ) as μ E ( A ) and view μ E as a Radon measure μ E on Ω . The measure μ E is called the H -perimeter measure of E . By the Riesz representation theorem, there exists a Borel function ν E : Ω R 2 n such that ν E = 1 μ E a.e. and the following formula

Ω V , ν E d μ E = Ω div H V d z d t

holds for any V C c 1 ( Ω ; R 2 n ) . We call the vector function ν E the horizontal inner normal of E in Ω .

Definition 2.2

A set E with locally finite perimeter is considered to be H-perimeter minimizing in U if:

(2.1) P H ( E , U ) P H ( F , U ) ,

for any set F H n such that the symmetric difference E F is a compact subset of U .

Let Y 1 be the vector field defined earlier, we define the closure of the cube Q r relative to the direction Y 1 as follows:

Q ¯ r Y 1 , + = { ( z , t ) H n : r < y 1 r , x 1 < r , t < r 2 , and x i , y i < r for i = 2 , , n } .

Definition 2.3

[12] (strongly perimeter minimizing) A set E with locally finite perimeter is considered to be strongly H-perimeter minimizing in some neighborhood U of 0, if for any Q r U ,

(2.2) P H ( E , Q r ) P H ( F , Q r ) ,

for any set F H n such that ( E F ) Q ¯ r is a compact subset of Q ¯ r Y 1 , + .

2.1 Excess and reduced boundary

Let E be a subset of H n with locally finite H -perimeter, we call 0 H n a point of the H -reduced boundary of E , denoted by 0 * E , if μ E ( B r ) > 0 for all r > 0 ,

lim r 0 1 μ E ( B r ) B r ν E d μ E = ν E ( 0 ) ,

and ν E ( 0 ) = 1 .

Take any p H n , r > 0 , and v S 2 n . The v-directional horizontal excess of E in B r ( p ) is

(2.3) Exc ( E , B r ( p ) , v ) = 1 r 2 n + 1 B r ( p ) ν E ( p ) v d μ E .

The horizontal excess of E in B r ( p ) is the minimum of all possible directional horizontal excesses of E :

(2.4) Exc ( E , B r ( p ) ) = min v S 2 n Exc ( E , B r ( p ) , v ) .

2.2 Intrinsic Lipschitz graph

Recall W = { ( z , t ) H n , x 1 = 0 } = R 2 n is the vertical hyperplane. For any function φ : W R , we define the intrinsic graph of φ along X 1 to be

(2.5) gr ( φ ) = { ( z + φ ( z , t ) e 1 , t + 2 y 1 φ ( z , t ) ) : ( z , t ) W } ,

and the intrinsic epigraph of φ along X 1 to be

(2.6) E φ = { ( z + s e 1 , t + 2 y 1 s ) : ( z , t ) W , s > φ ( z , t ) } ,

where z = ( x 1 , , x n , y 1 , , y n ) C n = R 2 n and e 1 = ( 1 , 0 , , 0 ) R 2 n .

Moreover, taking any point w = ( z , t ) W , we use the notation

(2.7) w * φ ( w ) ( z + φ ( w ) e 1 , t + 2 y 1 φ ( w ) ) .

Let V = ( e 1 , 0 ) H n , for any p H n , set e 1 ( p ) = p , V V H n to denote the projection of p on to e 1 , and let e 1 ( p ) be the point such that

(2.8) p = e 1 ( p ) * e 1 ( p ) .

Then the cone with vertex 0 H n , axis e 1 and aperture α ( 0 , ] is the set

(2.9) C ( 0 , v , α ) = { p H n : v ( p ) < α v ( p ) } .

Now, for a cone with vertex p instead of 0, we define the set as C ( p , v , α ) = p * C ( 0 , v , α ) .

Definition 2.4

Let D W be an open set. A continuous function φ : D R is a L-intrinsic Lipschitz function with L [ 0 , ) , if for any p gr ( φ ) , there holds

gr ( φ ) C ( p , v , 1 L ) = .

The gradient of intrinsic Lipschitz function is called intrinsic gradient.

Definition 2.5

Let D W be an open set, for any function φ Lip l o c ( D ) , we define the intrinsic gradient φ φ to be

(2.10) φ φ = ( X 2 φ , , X n φ , B φ , Y 2 φ , , Y n φ ) ,

where B is the Burgers’ operator,

(2.11) B φ = φ y 1 4 φ φ t .

If φ C ( D ) is a continuous function, we say that the intrinsic gradient φ φ exists in the sense of distributions if X i φ , B φ , Y i φ , i = 2 , , n exists in the sense of distributions. Then φ φ L loc ( D ; R 2 n 1 ) .

Monti [11] showed that the boundary of a set of minimizing H parameters E can be approximated with a L -Lipschitz graph. Let S 2 n + 1 denote the ( 2 n + 1 ) dimensional spherical Hausdorff metric associated with the Carnot-Carathedory distance. Then there is an intrinsic Lipschitz function φ such that the measure of the symmetric difference of gr ( φ ) and E is bounded by the excess, as Theorem 2.6 shows.

Theorem 2.6

(Lipschitz approximation) [11, Theorem 1.1] Let n 2 . For L > 0 , there exists some constant k > 1 such that for any H-perimeter minimizing set E in B k r , with 0 E and r > 0 , there exists an L-intrinsic Lipschitz function φ : W R such that

(2.12) S 2 n + 1 ( ( gr ( φ ) E ) B r ) c ( L , n ) ( k r ) Q 1 Exc ( E , B k r , X 1 ) ,

where c is some positive constant which depends on L and n.

2.3 Contact flow and first variation

Taking any bounded open set Ω H n , a contact map defined on Ω is a diffeomorphism Ψ such that the differential map Ψ * preserves the horizontal subbundle. That is, for any p Ω , Ψ * ( H p ) H Ψ ( p ) . Then a one-parameter flow ( Ψ s ) s R of H n is a contact flow if each Ψ s is a contact map. For more information on contact flows, see the study by Korányi and Reimann [9].

Take any generating function ψ C ( H n ) , let V ψ be the vector field in H n of the form

(2.13) V ψ = j = 1 n ( Y j ψ ) X j ( X j ψ ) Y j 4 ψ T .

Then there is a corresponding contact flow Ψ : [ δ , δ ] × Ω H n , such that for any s [ δ , δ ] , and p Ω , the following relation holds:

(2.14) Ψ ( s , p ) = V ψ ( Ψ ( s , p ) ) ,

(2.15) Ψ ( 0 , p ) = p .

We call the flow generated by some generating function ψ C ( H n ) .

Monti [10] showed the following first variation formula for a contact flow Ψ .

Theorem 2.7

[10, Theorem 3.18] Let Ω be some bounded open set in H n , and Ψ : [ δ , δ ] × Ω H n be a contact flow generated by some smooth function ψ C ( H n ) . Then there exists some positive constant C depending on ψ and Ω such that for any set E H n with a finite perimeter in Ω , we have

(2.16) P H ( Ψ s ( E ) , Ψ s ( Ω ) ) P H ( E , Ω ) + s Ω { 4 ( n + 1 ) T ψ + L ψ ( ν E ) } d μ E C P H ( E , Ω ) s 2 ,

for any s [ δ , δ ] , where L ψ : H p R is the real quadratic form,

(2.17) L ψ j = 1 n x j X j + y j Y j = i , j = 1 n x i x j X j Y i ψ + x j y i ( Y i Y j ψ X j X i ψ ) y i y j Y j X i ψ .

3 Proof of Theorem A

 We need to recall an approximation result before the proof of theorem A. Let E H n be a H -perimeter minimizing set in some neighborhood of 0, with 0 * E , ν E ( 0 ) = X 1 . Since 0 * E , there exists a sequence of real numbers r h 0 + such that

Exc ( E , Q r h ) < 1 h .

For any real number λ > 0 , let δ λ ( z , t ) = ( λ z , λ 2 t ) be the dilation defined on H n . Then the rescaled sets E h = δ 1 r h ( E ) satisfy the following properties:

  1. 0 * E h , and ν E h ( 0 ) = X 1 .

  2. Each set E h is H -perimeter minimizing.

  3. Exc ( E h , Q 1 ) < 1 h since excess is dilation invariant.

Let η h = Exc ( E h , Q 1 ) denote the square root of the excess of E h . Pick some small number σ such that 0 < σ < 1 k , where k is the geometric constant defined in Theorem 2.6. Then by Theorem 2.6, there exists a L -intrinsic Lipschitz function φ h : W R such that

(3.1) S Q 1 ( ( gr ( φ h ) E h ) B σ ) S Q 1 ( ( gr ( φ h ) E h ) B 1 k ) c ( L , n , σ ) ( 1 k k ) n + 1 Exc ( E h , B 1 ) = c 0 η h 2 ,

where c 0 = c ( L , n , σ ) .

By using this inequality and a Poincaré-type inequality proved in [5], Monti showed that there exists a subsequence of ( φ h η h ) h N that weakly converges to some function φ in the L 2 sense.

Theorem 3.1

[12, Theorem 2.5] Assume n 2 , following from the aforementioned construction, let φ h be the L-intrinsic function associated with E h . Then there exists an open neighborhood D W of 0, real constants φ h ¯ , and a selection of indices k h k such that as k , φ h k φ h k ¯ η h weakly converge to some function φ W H 1 , 2 ( D ) . Moreover, the intrinsic gradient φ h k φ h k also converges:

(3.2) φ h k φ h k η h k H φ weakly i n L 2 ( D ; R 2 n 1 ) ,

where

(3.3) H φ = X 2 φ , , X n φ , φ y 1 , Y 2 φ , , Y n φ .

Remark

In the proof of theorem 2.1, Monti also showed that

(3.4) φ h 0 and φ h ¯ 0 strongly in L 2 ( D ) ,

and the following estimate of intrinsic gradient

(3.5) D 1 φ h φ h 2 c 0 η h 2 ,

where c 0 is the constant in (3.1).

Without loss of generality, we can assume that D 1 = { ( z , t ) Q 1 : x 1 = 0 } D , then the limit function φ is defined on the whole D 1 .

Proof of Theorem A

The proof is a revised version of Monti’s proof of Theorem 3.2 in [12], where he used contact flow and first variation formula to obtain the final result. He assumed that the generating function ψ C ( H n ) for the contact flow Ψ : [ δ , δ ] × D 1 H n is of the form

ψ = α + x 1 β + 1 2 x 1 2 γ ,

where α , β , γ are smooth functions in H n such that

X 1 α = X 1 β = X 1 γ = 0 in Q 1 2 .

He further assumes that β , γ are compactly supported in Q 1 2 . We can identify two cases in Theorem A just as Monti did in ( 3.45 ) and ( 3.46 ) :

  1. If E is H -perimeter minimizing in Q 1 , we assume:

    (3.6) α C c ( Q 1 2 ) .

    Then by (2.13), the contact vector field V ψ vanishes outside Q 1 2 . We have E h Ψ s ( E h ) Q 1 for any s ( 0 , δ ] . Hence by the definition of H -perimeter minimizing, P H ( E h , Q 1 ) P H ( Ψ s ( E h ) , Q 1 ) .

  2. If E is strongly H -perimeter minimizing in Q 1 , we will first define α 0 on the vertical hyperplane W :

    (3.7) α 0 ( y 1 , z 2 , , z n , t ) = 0 y 1 ϑ 0 ( s , z 2 , , z n , t ) d s , y 1 R , z 2 , , z n C ,

    where ϑ 0 C c ( D 1 2 ) and X 1 ϑ 0 = 0 . Moreover, letting Π : H n W denote the nonlinear projection along the cosets of X 1 given by Π ( z 1 , , z n , t ) = ( 0 + i y 1 , z 2 , , z n , t 2 x 1 y 1 ) , we define α ( z 1 , , z n , t ) α 0 Π ( z 1 , , z n , t ) . Then X 1 α = 0 in H n , α is supported in the y 1 cylinder C y 1 = { ( z , t ) H n : x 1 < 1 2 , z i < 1 2 , t < 1 4 , i = 2 , , n } and remains constant for y 1 1 2 . Then V ψ is supported in C y 1 and equals 4 α T for y 1 1 2 . Then we have E h Ψ s ( E h ) Q 1 ¯ Q ¯ 1 Y 1 , + for any s ( 0 , δ ] . Hence, by the definition of strongly H -perimeter minimizing, P H ( E h , Q 1 ) P H ( Ψ s ( E h ) , Q 1 ) .

With abuse of notation, we use ψ , α , β , γ to denote its restriction on the plane { x 1 = 0 } . We also use the notations:

f t = f t , f y 1 = f y 1 ,

for any smooth functions f defined on D .

Then Monti applied the first variation formula (2.16) to each rescaled set E h . By the minimality condition P H ( E h , Q 1 ) P H ( Ψ s ( E h ) , Q 1 ) and the weak convergence of φ h η h , Monti concluded ( 3.49 ) in [12]:

(3.8) Δ P h lim h 1 η h D { 4 ( n + 1 ) T ψ ( w * φ h ( w ) ) + L ψ ( ν E φ h ( w * φ h ( w ) ) ) } d w = 0 ,

where D denotes the unit disk D 1 on the vertical plane W , L ψ is the quadratic form (2.17) associated with the first variation, E φ h is the intrinsic epigraph of φ h , and ν E φ h is the horizontal inner normal of E φ h .

As shown from (3.51) to (3.53) in [12], Monti computed the first half of Δ P h as follows:

(3.9) lim h 1 η h D 4 ( n + 1 ) T ψ ( w * φ h ( w ) ) d w = D 4 ( n + 1 ) β t φ .

It remains to calculate the second half of Δ P h

(3.10) lim h 1 η h D L ψ ( ν E φ h ( w * φ h ( w ) ) ) d w ,

The error occurred when Monti was expanding the form L ψ ( ν E φ h ) , on the third line of ( 3.56 ) in [12]. The term x 1 X j γ should be γ instead of x 1 X 1 γ when j = 1 . This caused the final integral ( 3.62 ) in [12] to miss a term γ φ y 1 .

By Theorem 2.4 in [12], the horizontal normal ν E φ h = ( ν X 1 , , ν X n , ν Y 1 , ν Y n ) is of the following form:

(3.11) ν X 1 = 1 1 + φ h φ h 2 , ν Y 1 = B φ h 1 + φ h φ h 2 ,

(3.12) ν X i = X i φ h 1 + φ h φ h 2 , ν Y i = Y i φ h 1 + φ h φ h 2 ,

for 2 i n , where B is the Burgers operator.

Then Monti rewrote L ψ as terms that contain ν X 1 plus some quadratic form K ψ ( ν E φ h )

(3.13) L ψ ( ν E φ h ) = ν X 1 2 X 1 Y 1 ψ + i = 2 n ( ν X i ν X 1 X 1 Y i ψ + ν X 1 ν X i X i Y 1 ψ ) + i = 1 n ν X 1 ν Y i ( Y i Y 1 ψ X 1 X i ψ ) + K ψ ( ν E φ h ) .

Then he computed the derivatives of ψ in L ψ that associate with ν X 1 :

(3.14) X 1 Y 1 ψ = Y 1 X 1 ψ 4 T ψ = Y 1 β + x 1 Y 1 γ 4 α t + x 1 β t + 1 2 x 1 2 γ t , X 1 Y i ψ = Y i X 1 ψ = Y i β + x 1 Y i γ ( i 2 ) , Y 1 Y i ψ = Y i Y 1 α + x 1 Y i Y 1 β + 1 2 x 1 2 Y i Y i γ ( i 1 ) , X 1 X 1 ψ = X 1 ( β + x 1 γ ) = γ , X 1 X i ψ = X i X 1 ψ = X i β + x 1 X i γ ( i 2 ) , X i Y 1 ψ = X i Y 1 α + x 1 X i Y 1 β + 1 2 x 1 2 X i Y 1 γ ( i 2 ) .

By inserting the derivatives into (3.13), we obtain the corrected version of ( 3.56 ) ,

(3.15) L ψ ( ν E φ h ) = L 1 ( h ) + L 2 ( h ) + L 3 ( h ) + L 4 ( h ) + K ψ ( ν E φ h ) ,

where we define

(3.16) L 1 ( h ) Y 1 β + x 1 Y 1 γ 4 α t + x 1 β t + 1 2 x 1 2 γ t ν X 1 2 ,

(3.17) L 2 ( h ) j = 2 n Y 1 X j α + x 1 Y 1 X j β + 1 2 x 1 2 Y 1 X j γ + Y j β + x 1 Y j γ ν X 1 ν X j ,

(3.18) L 3 ( h ) j = 2 n Y j Y 1 α + x 1 Y j Y 1 β + 1 2 x 1 2 Y j Y 1 γ X j β x 1 X j γ ν X 1 ν Y j ,

(3.19) L 4 ( h ) Y 1 Y 1 α + x 1 Y 1 Y 1 β + 1 2 x 1 2 Y 1 Y 1 γ γ ν X 1 ν Y 1 .

Since φ h is L -intrinsic Lipschitz, we can assume that its intrinsic gradient φ h φ h must be bounded everywhere. Hence, there exists some large constant A > 0 such that

(3.20) K φ ( ν E φ h ) A φ h φ h 2 .

By (3.5), we have D φ h φ h 2 c 0 η h 2 for some positive constant c 0 , then we obtain

(3.21) lim h 1 η h D K ψ ( ν E φ h ( w * φ h ( w ) ) ) d w lim h 1 η h D A φ h φ h 2 d w lim h A η h c 0 η h 2 = 0 .

Next, we compute the limit of the integral of L 1 ( h ) , L 2 ( h ) , L 3 ( h ) , and L 4 ( h ) . Since X 1 β = 0 , X 1 Y 1 β = Y 1 X 1 β 4 T β = 4 T β , and noticing that x 1 = φ h , we have

(3.22) Y 1 β ( w * φ h ( w ) ) = Y 1 β ( w ) 4 x 1 T β ( w ) = β y 1 ( w ) 4 φ h β t ( w ) .

Similarly, one can show that

(3.23) Y 1 γ ( w * φ h ( w ) ) = γ y 1 4 φ h γ t ,

(3.24) Y 1 α ( w * φ h ( w ) ) = α y 1 4 φ h α t .

Let φ h ¯ be the real constant in Theorem 3.1, by (3.4), we have

(3.25) lim h D φ h 2 η h f = lim h D φ h 2 φ h ¯ 2 η h f = lim h D φ h φ h ¯ η h ( φ h + φ h ¯ ) f = 0

for any smooth function f C ( D ) .

Moreover, since α is compactly supported in t 1 2 , β and γ are compactly supported in Q 1 2 , we have

(3.26) D β y 1 = D γ y 1 = D α t = D β t = 0 .

Noticing that ν X 1 = 1 as h goes to infinity by (3.11). Then by Theorem 3.1, (3.22), (3.25), and (3.26), the limit of the integral of L 1 ( h ) becomes

(3.27) lim h 1 η h D L 1 ( h ) d w = lim h 1 η h D ( β y 1 4 φ h β t ) + φ h ( γ y 1 4 φ h γ t ) 4 α t + φ h β t + 1 2 φ h 2 γ t ν X 1 2 d w = lim h D ( 8 β t + γ y 1 ) φ h η h d w = lim h D ( 8 β t + γ y 1 ) φ h φ h ¯ η h d w = D ( γ y 1 8 β t ) φ d w .

Similarly, using the fact that lim h φ h = 0 , Theorem 3.1, (3.24), and (3.12), the limit of integral of L 2 ( h ) becomes,

(3.28) lim h 1 η h D L 2 ( h ) d w = lim h 1 η h D i = 2 n ( Y i β + φ h Y i γ + X i Y 1 α + φ h X i Y 1 β + 1 2 φ h 2 X i Y 1 γ ) ν X 1 ν X i d w = lim h D i = 2 n ( Y i β + X i Y 1 α ) ν X 1 ν X i η h d w = lim h D i = 2 n ( Y i β ) + X i ( α y 1 φ h α t ) X i φ h η h d w = D i = 2 n ( Y i β + X i α y 1 ) X i φ d w .

Furthermore, one can use the same logic to compute the limit of the integral of L 3 ( h )

(3.29) lim h 1 η h D L 3 ( h ) d w = lim h 1 η h D i = 2 n ( Y i Y 1 α + φ h Y i Y 1 β + 1 2 φ h 2 Y i Y i γ ( X i β + φ h X i γ ) ν X 1 ν X i d w = lim h D i = 2 n ( Y i Y 1 α X i β ) ν X 1 ν Y i η h d w = lim h D i = 2 n ( Y i ( α y 1 4 φ h α t ) X i β ) Y i φ h η h d w = lim h D i = 2 n ( Y i α y 1 X i β ) Y i φ h η h d w = D i = 2 n ( Y i α y i + X i β ) Y i φ d w .

Finally, we compute the limit of the integral of L 4 ( h ) , which is the special case of Monti’s computation in (3.61) in [12], where j = 1 :

(3.30) lim h 1 η h D L 4 ( h ) d w = lim h 1 η h D ( Y 1 Y 1 ψ X 1 X 1 ψ ) ν X 1 ν Y 1 d w = lim h D Y 1 2 α + φ h Y 1 2 β + 1 2 φ h 2 Y 1 2 γ γ ν X 1 ν Y 1 η h d w = lim h D ( Y 1 2 α γ ) B φ h η h d w = D ( Y 1 2 α γ ) φ y 1 d w . = D φ y 1 Y 1 2 α + γ φ y 1 d w .

We obtain an additional γ φ y 1 term in the integral (3.30) compared to Monti’s original integral ( 3.61 ) .

Combining together (3.9) and (3.27) to (3.30), (3.8) becomes:

(3.31) D ( 4 ( n 1 ) β t + γ y 1 ) φ φ y 1 Y 1 2 α + γ φ y 1 i = 2 n [ ( X i α y 1 + Y i β ) X i φ + ( Y i α y 1 X i β ) Y i φ ] d w = 0 .

Note that this is ( 3.62 ) in [12] with the extra γ φ y 1 term.

Setting α = β = 0 , we have

(3.32) 0 = D γ y 1 φ + γ φ y 1 d w = D ( γ φ ) y 1 d w ,

which gives empty information since γ φ is a compactly supported function on D . Therefore, D ( γ φ ) y 1 automatically equals zero. We no longer obtain claim (i) in Theorem 3.2 in [12] because we no longer obtain the formula: 0 = D γ y 1 φ d w = D γ φ y 1 d w .

By letting β = γ = 0 , using integration by parts, we obtain

(3.33) 0 = D 2 α y 1 2 φ y 1 + i = 2 n ( X i α y 1 X i φ + Y i α y 1 Y i φ ) d w = D α y 1 2 φ y 1 2 + i = 2 n ( X i 2 φ + Y i 2 φ ) d w = D α y 1 Δ 0 φ d w = D α y 1 Δ 0 φ d w .

If E is perimeter minimizing, this holds for any compactly supported function α , and we obtain the following differential equation:

y 1 Δ 0 φ = 0 .

If E is strongly perimeter minimizing, α y 1 = ϑ for any test function ϑ C c ( D 1 2 ) , then we have

0 = D ϑ Δ 0 φ d w .

Hence, φ W H 1 , 2 ( D ) solves the partial differential equation Δ 0 φ = 0 in the weak sense.□

Acknowledgments

This material is based upon work supported by the National Science Foundation (Award No. 2005609). The author would like to thank Simone Verzellesi and Robert Young for their time and advice during the preparation of this paper.

  1. Author contribution: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.

  2. Conflict of interest: The authors state no conflict of interest.

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Received: 2024-08-23
Revised: 2025-01-22
Accepted: 2025-02-07
Published Online: 2025-03-14

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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